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Class: X MM: First Pre board Examination (2017 - 2018) 80 Subject: Mathematics Date: 27/12/17 Time: 2 1 2 hrs Attempt all questions from section A and any 4 questions from section B. All working, including the rough work, must be clearly shown and must be done on the same sheet as the rest of the answer. Omission of essential working will result in the loss of marks. The intended marks for questions or parts of questions are given in bracket [ ] SEC-A [ 40 Marks] Attempt all the questions from this section Question1 [ ] 4 1 . Find the value of k If M2 6M +kI = 0 where I is the identity 1 2 matrix of order 2 2. [3] a) Given M= b) Three numbers are in continued proportion. If the middle number is 18 and the sum of first and last is 39, find the numbers. [3] c) A manufacturer marks an article for `8000. He sells it to the wholesaler at 25% discount. The wholesaler sells it to a retailer at 20% discount on M.P. If the retailer sells it at M.P and VAT charged is 8% at every stage. Find the VAT paid to the Government by (i) the wholesaler (ii) the retailer. [4] Question2 a) When the two polynomials x3 px2 + x + 6 and 2x3 x2 (p + 3)x 6 are divided by (x 3), the remainder is same. Find the value of p. [3] b) If A (-6,0) and B(0, 6), find the points of trisection of the line segment AB. [3] c) The sum of the third and the seventh terms of an A.P is 6 and their product is 8. Find the sum of first sixteen terms of A.P [4] Question3 1 a) Cards marked with numbers 3, 4, 5, .., 50 are placed in a box and mixed thoroughly. One card is taken at random from the box. Find the probability that number drawn on the card is (i) Divisible by 7 (ii) a number which is divisible by 5 and not by 10 (iii) perfect square number. [3] b) Sanjana opens a Recurring Deposit scheme and deposits` 800 per month for a period for 2 years. If the rate of interest is 9% p.a., find the amount payable at the end of 2 years . [3] c) Using graph paper and taking 1cm = 1 unit along both x-axis and y-axis. (i) Plot the points A (-4,4) and B(2,2). (ii) Reflect A and B in the origin to get the images A and B respectively. (iii) Write down the co-ordinates of A and B . (iv) Give the geometrical name for the figure ABAB . [4] Question 4 a) In the adjoining figure, POR = 80 , Find the measure of PSR and PQR. 2 2 b) Prove that : tan A tan B = [3] sin2 A sin2 B . cos 2 A cos 2 B [3] c) A toy is the shape of a right circular cylinder with cone on one side and a hemisphere on the other, all with same radius 5cm, The height of the cylinder is 13 cm. Find the surface area of the toy if the total length is 30 cm. [4] 2 Sec B [40 marks] [Attempt any 5 Questions ] Question 5 a) Solve the quadratic equation 5x( x+2) = 3. Give your up to 3 significant digits. [3] b) Prove that : ( 1+tanA . tanB )2 [3] + ( tanA tanB )2 = sec 2 A sec 2 B . c) Arjun sold a certain no. of shares of ` 100 paying 10% dividend at a discount of 25% and invested the proceeds in `100 shares paying 16% dividend, quoted at `80 and thus increased his income by ` 2000 . Find the no. of shares he sold. [4] Question 6 a) If a, b, c are in continued proportion prove that (a2 b2)(b2+c2) = (b2 c2) ( a2 + b2) [3] b) In the given figure, AC bisects BAD and AB = 16 cm , AC= 12 cm, Ad = 9 cm. Prove area of ABC that ABC ~ ACD (ii) Find areaof ACD [3] c) From the top of a tower 60 m high, the angles of depression of the top and bottom of a building are observed to be 30 and 60 respectively. Find the height of the building and the distance between them to the nearest metre. [4] Question 7 a) Solve the following linear in equation and represent the solution set on the number line: 4x 19< 3x 2 2 + x , x R 5 5 [3] b) How many terms of A.P 9, 17, 25 must be taken to give a sum of 636? [3] c) The mean wages of 50 workers is ` 84 find the value of x and y. [4] Wages in ` No. of workers 50 60 70 80 90 100 110 2 x y 12 10 5 8 3 Question 8 a) Chord AB and CD of circle at a point P. Prove that PA PB = PC PD. If BP= 3cm, CD = 10 cm, and PA = 8 cm, Find the length of CP. b) If A = [ ] 4 5 , 3 4 [3] B= [] 7 6 and AX = B find the order of matrix X (ii) the matrix X. c) In the figure equation AB is x (i) 3 y +1 = 0 and the equation of AC is x y 2 = 0. Write down the angles that AB and Ac make with the positive direction of x axis. (ii) find BAC (iii) Find the coordinates of points A and B. [4] Question 9 a) A hemisphere is surmounted on a conical block of wood. The diameter of their base is 6cm each and the slant height of the cone is 5cm. Find (i) the height of the cone (ii) the volume of the solid to the nearest cm3 .[ use =3.14] [3] b) In the given figure , Chord AB is parallel to the diameter AB, If ABC= 25 , find (i) BCD (ii) AEB [3] (iii) CED 4 c) From the top of a building 20m high, the angle of elevation of the top of the monument is 45 , and the angle of depression of its foot is 15 . Find the height of the monument correct to three significant digits. [4] Question 10 a) Amitab invests `42,000 in `100 tea shares of a company at a premium 20% from the market. The company pays 12% dividend annually. He sold 50% of the shares when the price rises to `150. Find (i) The number of shares bought by him. (ii) His annual income from the remaining tea shares. (iii) The money he received by selling the shares. (iv) The profit made by him on this transaction. b) The following distribution represents the height of 160 students of a school. Height 140in cm 145 No of 12 students 145150 20 150155 30 155160 38 160165 24 165170 16 170175 12 175180 8 Draw an ogive for the given distribution taking 2cm = 5cm height on one axis and 2cm = 20 students on the other axis. Using the graph Determine: (i) The median (ii) The inter quartile range (iii) The number of students whose height is above 172cm [6] Question11 a) A side of an equilateral triangle is 20cm long. A second equilateral triangle is formed by joining the midpoints of the sides of the first triangle. The process is continued as shown in the adjoin figure. Find the perimeter of 6th inscribed triangle. [3] b) Solve using the properties of proportion: x 3 +27 x 9 x2 +27 = 63 62 [3] c) A car covers a distance of 400km at a certain speed. Had the speed been 12km/h more, the time taken for the journey would have been 1 hour 40 minutes less. Find 5 the original speed of the car. [4] 6
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